Chapter4.Rmd
summary(cars)## speed dist
## Min. : 4.0 Min. : 2.00
## 1st Qu.:12.0 1st Qu.: 26.00
## Median :15.0 Median : 36.00
## Mean :15.4 Mean : 42.98
## 3rd Qu.:19.0 3rd Qu.: 56.00
## Max. :25.0 Max. :120.00
Chapter 4 - Clustering and classification
First we load the “Boston” data from MASS package and explore it.
library(MASS)
data("Boston")
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00names(Boston)## [1] "crim" "zn" "indus" "chas" "nox" "rm" "age"
## [8] "dis" "rad" "tax" "ptratio" "black" "lstat" "medv"The “Boston”-data is comprised of 506 observations of 14 different variables. The “Names()”-function provides the names of the variables included. The function “Summary()” produces the distributions of each variable.
Now we explore the relationship between variables.
library(corrplot)## corrplot 0.84 loadedlibrary(tidyverse)## ── Attaching packages ───────────────────────────────────────────────────────────────────────── tidyverse 1.2.1 ──## ✔ ggplot2 3.1.0 ✔ purrr 0.2.5
## ✔ tibble 1.4.2 ✔ dplyr 0.7.7
## ✔ tidyr 0.8.2 ✔ stringr 1.3.1
## ✔ readr 1.2.1 ✔ forcats 0.3.0## ── Conflicts ──────────────────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ✖ dplyr::select() masks MASS::select()cor_matrix<-cor(Boston) %>% round(digits=2)
corrplot(cor_matrix, type="upper", cl.pos = "b",tl.pos = "d",tl.cex = 0.6 )
In the graph, the blue colour represents positive correlation between variables while the red rpresents the negatives. The darker the colour and the bigger the ball, the stronger the correlation is. For example, the big blue ball between “rad” and “tax” tells that there is a strong correlation between access to highways and property tax rate.
Now we’ll standardise the data
boston_scaled <- scale(Boston)
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865Now all the observations are using the same scale. Making assumptions about the summary for example is meaningful.
Then we must turn “boston_scaled” back into a data.frame
boston_scaled <- as.data.frame(boston_scaled)These lines give us the cathegorical variable “crime”
bins <- quantile(boston_scaled$crim)
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))And these replace the old variable “crim” with the cathegorical crime
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)Then we divide the data to train with 80% of the data and test with 20% of the data.
n <- nrow(boston_scaled)
ind <- sample(n, size = n * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]- Because the data has now been cathegorised into separate sets for training and testing, we can use the training-part for linear discriminant analysis, where crime rate will be predicted by all of the other variables.
lda.fit <- lda(crime ~ ., data = train)
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
classes <- as.numeric(train$crime)
plot(lda.fit, dimen = 2, col=classes)
lda.arrows(lda.fit, myscale = 2)
Looking at the bi-plot, it’s clear that the “rad”-variable acts (on its own) as a predictor of “high crime rate” in the Boston data. The other 12 variables are associated with low, medium low and medium high rates of crime. The grouping based on these 12 variables is quite vague and it is difficult to see whether any of the variables can accurately/adequately sort the associated observations.
crime_cat<-test$crime
test<-dplyr::select(test, -crime)
lda.pred<-predict(lda.fit, newdata = test)
table(correct = crime_cat, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 17 8 1 0
## med_low 8 16 1 0
## med_high 1 11 15 1
## high 0 0 0 23The amount of correct cases and of predicted cases for each of the cathegories (low, med_low, med_high, high) varies between every sample matrix. The change is to be expected as the sets we have created have been randomly classified. There is much less variation between predictions for the “high” class than there is in the predictions of the others.
- Before we practice K-means clustering, we should reload the Boston data, scale it, and afterwards measure the distances between the observations.
data(Boston)
boston_scaled1<-as.data.frame(scale(Boston))
dist_eu<-dist(boston_scaled1)
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970head(boston_scaled1)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv
## 1 -1.0744990 0.1595278
## 2 -0.4919525 -0.1014239
## 3 -1.2075324 1.3229375
## 4 -1.3601708 1.1815886
## 5 -1.0254866 1.4860323
## 6 -1.0422909 0.6705582The scaled Boston data will now be used for K-means clustering. It isn’t trivial (in many cases) to investigate on the number of clusters that can classify the data. Therefore, we need to first randomize the usage of a certain number of clusters.
First let’s start with a random number cluster. Let us choose k=4 and apply k-means on the data.
kmm = kmeans(boston_scaled1,6,nstart = 50 ,iter.max = 15) The elbow method is one good technique using which we can estimate the number of clusters.
library(ggplot2)
set.seed(1234)
k.max <- 15
data <- boston_scaled1
wss <- sapply(1:k.max,
function(k){kmeans(data, k)$tot.withinss})
qplot(1:k.max, wss, geom = c("point", "line"), span = 0.2,
xlab="Number of clusters K",
ylab="Total within-clusters sum of squares")## Warning: Ignoring unknown parameters: span
## Warning: Ignoring unknown parameters: span
The elbow plot seems to indicate that we may not find more than two clear clusters but it’s good to confirm predictions using another method because there is no shortage of methods for analyses like these. Let’s try the “NbClust”- package.
library(NbClust)
nb <- NbClust(boston_scaled1, diss=NULL, distance = "euclidean",
min.nc=2, max.nc=5, method = "kmeans",
index = "all", alphaBeale = 0.1)
## *** : The Hubert index is a graphical method of determining the number of clusters.
## In the plot of Hubert index, we seek a significant knee that corresponds to a
## significant increase of the value of the measure i.e the significant peak in Hubert
## index second differences plot.
## 
## *** : The D index is a graphical method of determining the number of clusters.
## In the plot of D index, we seek a significant knee (the significant peak in Dindex
## second differences plot) that corresponds to a significant increase of the value of
## the measure.
##
## *******************************************************************
## * Among all indices:
## * 12 proposed 2 as the best number of clusters
## * 6 proposed 3 as the best number of clusters
## * 3 proposed 4 as the best number of clusters
## * 3 proposed 5 as the best number of clusters
##
## ***** Conclusion *****
##
## * According to the majority rule, the best number of clusters is 2
##
##
## *******************************************************************hist(nb$Best.nc[1,], breaks = max(na.omit(nb$Best.nc[1,])))
Now, it’s easier to see that the data is described better with two clusters. With that, we should run the k-means algorithm again.
km_final = kmeans(boston_scaled1, centers = 2)
pairs(boston_scaled1[3:9], col=km_final$cluster)
The clusters in the above plot are divided into two groups and outlined using the colors red and black. Some of the pairs are better grouped than other ones in the plot. One of the important observations can be made with the “chas”-variable where the observations in all of the pairs formed by it are wrongly clustered. Still, clusters formed by the “rad” variable are better separated.
Bonus:
Now, we will use a randomly selected cluster number (k=6) and perform LDA. We shall follow the the basic steps of scaling and distance calculation. Afterwards, we will find out how the biplot looks (of the whole data set) as we try to group it into six different categories.
boston_scaled2<-as.data.frame(scale(Boston))
head(boston_scaled2)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv
## 1 -1.0744990 0.1595278
## 2 -0.4919525 -0.1014239
## 3 -1.2075324 1.3229375
## 4 -1.3601708 1.1815886
## 5 -1.0254866 1.4860323
## 6 -1.0422909 0.6705582set.seed(1234)
km_bs2<-kmeans(dist_eu, centers = 6)
head(km_bs2)## $cluster
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
## 6 6 6 6 6 6 6 4 4 6 4 6 6 4 4 4 6 4
## 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
## 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
## 4 6 6 1 1 6 6 6 6 6 6 4 4 6 6 6 6 6
## 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
## 1 1 1 1 6 6 6 6 6 6 1 1 1 6 6 6 6 6
## 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 6 6 6 6 4 4 4 6 6 6 6 6 6 6 6 6 6 6
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
## 6 6 6 6 4 6 6 3 3 6 4 4 4 4 4 4 4 4
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
## 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 4 4
## 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
## 2 2 2 2 4 4 4 4 2 2 2 2 2 2 2 2 5 2
## 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
## 5 5 5 5 5 2 2 2 5 2 5 5 5 3 4 2 3 3
## 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 3 3 4 4 3 4 4 4 4 4 4 4 4 6 6 6 6 6
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
## 3 6 6 6 4 6 3 6 6 1 1 1 1 1 1 1 1 1
## 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
## 1 1 1 1 1 1 1 6 6 4 3 3 3 3 3 6 4 6
## 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234
## 3 4 3 3 3 3 3 6 3 3 3 6 3 6 4 6 3 3
## 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
## 3 4 3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
## 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
## 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3
## 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288
## 6 6 6 3 3 6 3 3 6 6 3 6 3 1 1 1 1 6
## 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306
## 6 1 1 1 1 6 6 6 6 6 1 1 1 6 6 6 6 6
## 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324
## 6 6 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 4
## 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342
## 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 1
## 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
## 6 6 1 6 6 1 1 1 6 1 1 1 1 1 5 5 5 2
## 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
## 2 2 2 5 5 5 2 5 5 5 5 2 5 5 5 2 2 2
## 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396
## 2 2 5 2 2 2 5 5 5 5 5 2 2 2 2 2 2 2
## 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414
## 2 2 5 2 5 2 2 2 5 5 5 2 2 2 5 5 5 5
## 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432
## 5 5 5 5 5 5 2 2 2 5 5 5 5 5 5 5 2 5
## 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
## 2 2 5 5 5 5 5 2 2 2 2 2 2 5 2 2 2 2
## 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468
## 5 2 2 2 5 5 5 5 2 2 2 2 2 2 2 2 5 2
## 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504
## 2 2 2 2 5 2 2 4 4 4 4 4 4 4 4 4 4 4
## 505 506
## 4 4
##
## $centers
## 1 2 3 4 5 6 7 8
## 1 3.989644 4.556297 4.216580 3.977768 4.072917 4.089124 4.248097 4.850842
## 2 5.635860 5.015922 5.810840 6.243394 6.219170 5.797678 5.007134 5.045384
## 3 4.115845 4.367908 3.914214 4.415556 4.277226 4.521520 4.462950 4.591271
## 4 3.322873 2.419949 3.555546 3.901515 3.917887 3.259432 2.872770 3.145128
## 5 7.347551 6.917018 7.668887 8.029178 8.008552 7.594726 6.746593 6.719663
## 6 2.552963 2.191615 2.547985 2.481257 2.636215 2.126035 2.463163 3.260455
## 9 10 11 12 13 14 15 16
## 1 6.023239 4.730761 5.085827 4.546510 4.256435 5.022876 5.346690 5.064489
## 2 5.275376 5.034209 5.024306 5.008656 5.191566 4.483294 4.239085 4.500312
## 3 5.861940 4.897185 5.053174 4.724166 4.806196 5.041859 5.037049 5.123212
## 4 3.802664 3.078288 3.240106 2.958259 3.171899 2.294340 2.083575 2.330854
## 5 6.637733 6.672022 6.656885 6.725987 6.831291 6.522547 6.271019 6.514117
## 6 4.420281 3.050288 3.460627 2.800817 2.548810 2.498158 2.847417 2.519525
## 17 18 19 20 21 22 23 24
## 1 4.839017 5.494447 5.410418 5.409972 6.301722 5.559564 5.849309 6.119225
## 2 4.932293 4.118743 4.699219 4.231080 4.140866 4.107180 4.064205 4.123917
## 3 5.188031 5.159236 5.556072 5.185846 5.777782 5.080391 5.333884 5.590056
## 4 2.926372 2.006490 2.829410 2.073005 2.478201 1.996680 2.200197 2.363123
## 5 6.848068 6.136584 6.246617 6.243108 5.944153 6.154788 6.070741 6.047729
## 6 2.477710 2.972866 3.035484 2.815474 3.883785 3.047242 3.413583 3.706860
## 25 26 27 28 29 30 31 32
## 1 5.776453 5.909337 5.645826 5.792679 5.463721 5.291108 6.251560 5.790675
## 2 4.125236 4.229138 4.173627 4.154204 4.209655 4.253839 4.190000 4.132815
## 3 5.355804 5.648491 5.331440 5.423799 4.956368 4.757484 5.818922 5.290357
## 4 2.152358 2.399737 2.130704 2.313981 2.194705 2.226566 2.563711 2.185395
## 5 6.124494 5.850285 6.119300 5.848617 6.281408 6.323214 5.926407 6.111869
## 6 3.317906 3.492059 3.174430 3.407128 3.072595 2.917289 3.883866 3.341829
## 33 34 35 36 37 38 39 40
## 1 6.577978 6.113220 6.286376 5.003424 4.975828 4.672432 4.567581 2.437628
## 2 4.543962 4.083659 4.244460 4.504907 4.504214 4.851253 4.995628 6.988766
## 3 6.168152 5.623677 5.717537 4.728770 4.767845 4.847078 4.787796 5.456166
## 4 3.295133 2.297075 2.760000 2.036929 2.052171 2.484306 2.720323 5.019539
## 5 5.702765 5.899302 5.635547 6.472198 6.389736 6.751702 6.860251 8.586641
## 6 4.437132 3.655982 4.014459 2.356225 2.324457 2.085109 2.100542 3.469146
## 41 42 43 44 45 46 47 48
## 1 2.550075 4.003727 4.101822 4.105741 4.205545 4.493890 4.535762 5.127133
## 2 7.347068 6.266063 6.066725 6.010008 5.405380 5.401641 5.323536 4.969192
## 3 5.469978 4.947438 5.088076 5.079073 4.810066 5.080046 5.057556 5.071546
## 4 5.447668 4.089165 3.798403 3.739937 2.899376 2.938610 2.889613 2.590240
## 5 8.944151 7.985202 7.757987 7.727039 7.185569 7.146905 7.056687 6.715830
## 6 3.806323 2.464106 2.277093 2.245886 1.890630 2.124631 2.187826 2.870058
## 49 50 51 52 53 54 55 56
## 1 6.354068 4.788056 3.687296 3.635669 3.127248 3.365371 3.768251 2.573205
## 2 5.425549 5.185207 5.660587 5.607665 6.251113 6.093472 6.228695 8.172561
## 3 6.206363 5.152123 5.037840 4.798345 4.875844 5.098446 6.488456 6.248718
## 4 3.649819 2.729138 3.311553 3.194530 3.999725 3.806354 4.738612 6.360064
## 5 6.857506 6.903606 7.327416 7.333311 7.954340 7.750576 7.846534 9.656599
## 6 4.374890 2.484987 2.216591 2.155992 2.202966 2.192392 4.011314 4.620562
## 57 58 59 60 61 62 63 64
## 1 2.435993 2.586150 3.455264 3.762957 4.163099 4.634886 3.692547 3.384146
## 2 7.625711 8.169946 5.852530 5.357849 5.193164 4.988228 5.388941 5.749506
## 3 6.314081 6.255593 5.396948 5.291164 5.423722 5.411219 4.967901 5.080093
## 4 5.728217 6.380166 3.850185 3.257986 3.128914 3.007457 3.306577 3.782189
## 5 9.093653 9.562610 7.619436 7.182935 6.981514 6.757826 7.282673 7.580278
## 6 4.138053 4.788808 2.369128 2.238707 2.577586 3.001430 2.348247 2.423646
## 65 66 67 68 69 70 71 72
## 1 3.572872 2.365488 2.854471 3.828722 4.245479 3.861492 4.311560 4.482338
## 2 6.803815 7.202476 6.871459 5.702754 5.382884 5.506994 5.679697 5.332385
## 3 5.070095 5.920134 6.075955 5.331567 5.513971 5.232351 5.172823 5.258275
## 4 4.571372 5.363377 5.039822 3.540862 3.214330 3.286807 3.670384 3.233668
## 5 8.474576 8.706790 8.325184 7.481800 7.133931 7.317062 7.509722 7.139492
## 6 3.099810 3.755717 3.762213 2.164752 2.377335 2.067120 2.395258 2.279879
## 73 74 75 76 77 78 79 80
## 1 4.423047 4.374894 4.565665 4.571969 4.993178 4.715682 4.645174 4.703818
## 2 5.670343 5.628241 5.253844 4.504533 4.080734 4.392771 4.369389 4.613917
## 3 5.331200 5.256931 5.138078 4.739219 4.676271 4.796607 4.781211 5.037161
## 4 3.633211 3.612120 3.532216 2.554765 2.115305 2.435091 2.406743 2.698489
## 5 7.496987 7.426016 7.172186 6.550720 6.164051 6.445026 6.421550 6.621032
## 6 2.419650 2.395619 2.582255 2.208724 2.569263 2.284706 2.281898 2.322846
## 81 82 83 84 85 86 87 88
## 1 3.228722 3.647245 3.352955 3.510806 4.214393 4.216222 4.470996 4.539526
## 2 5.833680 5.345002 5.671433 5.439196 5.298921 5.378578 5.181960 5.119600
## 3 4.611003 4.507704 4.824876 4.799211 4.493640 4.250144 4.736784 4.543074
## 4 3.593223 2.978716 3.362220 3.049922 2.723815 2.845202 2.603436 2.477017
## 5 7.679009 7.247432 7.495607 7.269305 7.155841 7.257393 6.986917 6.981295
## 6 2.077143 2.128207 1.941076 1.906614 1.839655 1.960338 2.004413 2.011357
## 89 90 91 92 93 94 95 96
## 1 4.787782 4.431266 4.656816 4.754404 4.105175 4.001644 4.521147 4.347592
## 2 5.260876 5.460377 5.023131 4.997348 4.772459 5.169144 4.479851 5.437460
## 3 4.086432 3.906040 4.252949 4.267811 4.510455 4.766874 4.611235 4.143402
## 4 2.894806 3.107514 2.443156 2.419531 2.751041 3.222217 2.448424 2.937590
## 5 7.155386 7.332941 6.878036 6.862533 6.774331 7.083584 6.505216 7.188899
## 6 2.659893 2.401307 2.216247 2.322228 2.415933 2.468939 2.704529 2.182013
## 97 98 99 100 101 102 103 104
## 1 4.715695 4.926015 4.640776 4.444450 5.282563 5.152315 6.572203 5.683531
## 2 5.051112 6.412041 6.764467 5.837829 4.205067 4.305317 5.036975 3.815398
## 3 4.508034 3.931800 4.158120 3.883013 4.489959 4.495810 6.000236 5.001289
## 4 2.400338 4.464342 4.810546 3.603211 2.414290 2.493991 3.966604 1.972705
## 5 6.868421 8.203118 8.491487 7.681807 6.393702 6.492260 5.811685 5.987192
## 6 2.261612 3.692387 3.630574 2.739459 2.966666 2.843788 4.692793 3.172095
## 105 106 107 108 109 110 111 112
## 1 5.724543 6.103378 6.104999 5.770964 5.777602 5.809233 5.260548 5.174233
## 2 3.840428 3.789013 3.785737 3.801714 3.887880 3.778476 4.058712 3.895295
## 3 4.956142 5.253274 5.307506 4.990050 4.894350 5.027813 4.918935 4.183520
## 4 2.022137 2.164155 2.196416 2.023648 2.172750 2.042266 2.184470 2.238546
## 5 6.014221 5.881611 5.856170 5.944714 6.084653 5.930743 6.185964 6.069764
## 6 3.223448 3.618628 3.631351 3.259679 3.351457 3.335422 2.743048 2.868558
## 113 114 115 116 117 118 119 120
## 1 5.795991 5.778504 5.457332 5.719642 5.202323 5.373780 5.497713 5.357061
## 2 3.601847 3.596459 3.722199 3.590667 3.748787 3.744213 3.671311 3.776235
## 3 4.819193 4.774286 4.530960 4.818453 4.451377 4.589012 4.741708 4.798190
## 4 2.075218 2.106269 2.052459 2.053031 1.971584 1.988655 2.048517 2.045794
## 5 5.667421 5.680606 5.851170 5.493919 5.878062 5.869090 5.544023 5.809181
## 6 3.375466 3.392521 3.026294 3.301736 2.747953 2.915253 3.063432 2.869590
## 121 122 123 124 125 126 127 128
## 1 6.426088 6.565457 6.806695 7.305465 6.897705 6.650747 7.533959 7.099802
## 2 4.261019 4.162980 4.128457 4.247333 4.123875 4.163650 4.355189 3.161437
## 3 5.427653 5.424854 5.565025 6.069215 5.659762 5.427403 6.344535 5.999576
## 4 2.985164 2.938368 3.039074 3.475730 3.062302 2.973639 3.677551 2.962007
## 5 6.237519 6.139760 6.064278 6.001360 6.051491 6.152869 5.974765 5.445330
## 6 4.116775 4.252645 4.510005 5.058229 4.583383 4.343081 5.287399 4.735059
## 129 130 131 132 133 134 135 136
## 1 6.875876 7.154803 6.726119 6.686918 6.596530 6.821369 7.166802 6.870993
## 2 3.183589 3.174651 3.265601 3.234532 3.348169 3.162883 3.338913 3.134950
## 3 5.627423 6.120486 5.509731 5.516882 5.382470 5.779404 6.139554 5.674601
## 4 2.931268 3.021401 2.885217 2.835481 2.871725 2.806058 3.188140 2.903821
## 5 5.604137 5.425648 5.688097 5.652880 5.721190 5.497188 5.035538 5.535608
## 6 4.590233 4.791289 4.456575 4.402201 4.350142 4.479053 4.894039 4.582035
## 137 138 139 140 141 142 143 144
## 1 6.929533 6.859261 7.327005 7.036231 7.324311 8.387851 9.118028 8.148386
## 2 3.098547 3.195797 3.203702 3.121999 3.270140 4.203114 6.057048 4.491078
## 3 5.819167 5.632765 6.197877 5.809014 6.167568 7.327389 6.736697 6.601594
## 4 2.859060 2.928458 3.172007 2.969872 3.289011 4.421385 6.210476 4.719098
## 5 5.406565 5.610438 5.411571 5.483828 5.434774 5.732297 6.519423 5.705948
## 6 4.585322 4.572210 4.994596 4.721431 5.047669 6.196553 7.427033 6.200054
## 145 146 147 148 149 150 151 152
## 1 8.591160 8.420611 8.051529 8.485530 8.220414 7.812856 7.329573 7.603008
## 2 4.881773 4.956580 4.786268 4.850486 4.658853 4.298230 4.279168 4.390872
## 3 7.209721 6.881696 6.562636 7.091389 6.756783 6.281936 5.599987 6.029305
## 4 5.120965 5.221320 4.864032 5.058645 4.839584 4.425501 4.201583 4.325365
## 5 5.882920 5.395652 5.350043 5.860027 5.668932 5.539694 5.844264 5.733672
## 6 6.628290 6.634143 6.205886 6.539074 6.293195 5.851136 5.439102 5.651936
## 153 154 155 156 157 158 159 160
## 1 8.622664 7.648725 8.381752 8.889137 8.425203 6.168678 6.265700 7.165808
## 2 6.067192 4.403229 5.843835 6.536516 5.328492 5.218020 4.320602 4.558378
## 3 6.369979 6.052779 5.844207 6.703763 7.158351 4.234387 4.756091 5.368031
## 4 5.967163 4.417624 5.805631 6.566625 5.400744 4.276378 3.305864 4.351927
## 5 6.541819 5.446370 6.344801 6.334474 5.503366 6.942094 6.074535 6.178603
## 6 6.952748 5.752673 6.793409 7.450863 6.650053 4.652464 4.273222 5.383270
## 161 162 163 164 165 166 167 168
## 1 7.231434 6.383671 7.606315 7.711715 6.150603 6.283786 6.542030 6.239898
## 2 5.940257 6.088162 7.394089 7.646620 4.086328 4.360674 6.228342 4.356613
## 3 4.605274 4.398804 4.791045 4.937649 4.851661 4.926498 4.456582 5.095670
## 4 5.239834 5.208814 6.712875 6.998166 3.062315 3.452702 5.427142 3.432231
## 5 6.821006 7.745103 8.432676 8.695086 5.936568 5.663492 7.854149 5.576630
## 6 5.752473 5.237762 6.745173 6.961257 4.080738 4.385762 5.475612 4.315636
## 169 170 171 172 173 174 175 176
## 1 6.213385 6.133807 6.433082 6.312035 5.454412 4.962647 4.941506 4.309193
## 2 4.170734 4.100748 4.054129 4.025673 4.699883 4.808037 4.816591 5.496225
## 3 4.728033 4.677209 5.202442 5.015219 4.676351 4.091021 4.407110 4.153032
## 4 3.234454 3.142836 3.181914 3.077576 2.488721 2.512347 2.447970 3.326034
## 5 5.730715 5.794348 5.516106 5.715890 6.426784 6.668081 6.607389 7.275149
## 6 4.254128 4.150282 4.374444 4.230784 3.104965 2.654763 2.534686 2.342553
## 177 178 179 180 181 182 183 184
## 1 4.520962 4.655885 4.649899 4.604653 5.132343 4.814462 5.087352 5.054530
## 2 5.015596 4.976181 5.157347 5.945635 6.198433 5.587189 5.931557 5.543818
## 3 4.411335 4.095254 3.737891 3.879091 3.800673 4.190561 3.860567 4.061937
## 4 2.658053 2.616342 2.975519 3.642452 4.178165 3.143454 3.744498 3.144853
## 5 6.792977 6.830833 7.010189 7.720425 7.949502 7.318332 7.703218 7.337983
## 6 2.175426 2.365924 2.589714 2.810682 3.656914 2.701373 3.342620 2.979474
## 185 186 187 188 189 190 191 192
## 1 5.360158 4.777155 5.085672 3.049814 2.837691 2.892236 2.592917 2.487691
## 2 5.146074 5.258702 6.977902 6.090983 6.253786 6.421824 6.840347 6.509686
## 3 4.706625 4.276810 4.103340 4.369622 4.665779 4.360736 4.840840 4.814221
## 4 2.617485 2.667473 5.069928 4.220068 4.355866 4.608931 5.027524 4.620020
## 5 6.823279 6.988966 8.637738 7.764729 7.876360 8.097468 8.409214 8.135143
## 6 3.009695 2.461779 4.108449 2.947084 2.866184 3.171421 3.299804 2.939362
## 193 194 195 196 197 198 199 200
## 1 2.626002 2.323019 2.285881 3.495329 2.818135 2.825122 2.840680 2.451539
## 2 6.916387 7.350478 7.173909 8.592641 8.185358 7.919777 8.103044 7.965488
## 3 4.786382 5.315615 5.265629 5.625695 5.786297 5.784978 5.720384 6.255528
## 4 5.115810 5.261122 5.038340 6.858470 6.362933 6.100498 6.288002 6.311681
## 5 8.533158 8.873489 8.672208 10.009282 9.580192 9.208468 9.481059 9.444290
## 6 3.385297 3.423873 3.258558 5.286849 4.731058 4.582926 4.715512 4.639590
## 201 202 203 204 205 206 207
## 1 2.456624 2.615509 2.868195 3.538131 3.737246 4.677119 4.523642
## 2 7.969849 7.156198 8.271244 8.640337 8.822374 4.966944 4.612421
## 3 6.272606 5.771920 5.739854 5.855113 5.957838 4.945805 4.407545
## 4 6.329487 5.273529 6.554190 6.965429 7.174721 2.820461 2.266758
## 5 9.430213 8.587107 9.723775 10.042167 10.216692 6.837520 6.613955
## 6 4.653597 3.907488 4.866644 5.491799 5.693985 2.252863 1.996460
## 208 209 210 211 212 213 214 215
## 1 5.173003 6.173648 7.246427 6.672900 7.168488 6.358530 4.424781 5.994199
## 2 4.338029 5.951508 5.923358 5.785418 5.899356 5.931043 4.957469 5.447417
## 3 4.839209 4.297639 5.214301 4.486356 5.179802 4.522080 4.441468 6.128544
## 4 2.088170 4.501917 4.748828 4.456045 4.719108 4.521710 2.774437 3.942573
## 5 6.238923 7.056415 6.886396 6.882747 6.859678 7.017952 6.877368 6.804923
## 6 2.684155 4.559083 5.472929 4.942984 5.391736 4.669469 2.088834 4.075614
## 216 217 218 219 220 221 222 223
## 1 4.533967 6.461391 5.090497 6.968673 6.593520 6.320911 6.799719 6.130295
## 2 4.734048 5.887167 4.502200 5.707295 5.831456 6.068344 5.809474 6.083430
## 3 4.493721 4.333352 3.927803 4.474244 4.098632 3.855927 4.507648 3.793452
## 4 2.431159 4.642940 2.604000 4.619602 4.623502 4.781149 4.654611 4.752233
## 5 6.701716 6.923664 6.481639 6.712071 6.932306 7.216588 6.814993 7.226735
## 6 2.008161 4.833935 2.937343 5.253528 4.990749 4.858718 5.140070 4.702050
## 224 225 226 227 228 229 230 231
## 1 4.738585 5.386965 5.891695 5.165618 4.738937 4.794451 4.323631 4.757661
## 2 4.706017 6.297753 6.888318 5.848384 4.984604 6.583434 5.480585 4.442431
## 3 3.859178 3.856074 4.287951 3.761622 3.704484 4.327344 4.344291 4.345763
## 4 2.632171 4.857703 5.604564 4.291528 3.086868 5.025870 3.556647 2.180368
## 5 6.676587 8.061548 8.562013 7.691942 6.882629 8.267586 7.279603 6.313200
## 6 2.510992 4.285144 5.053107 3.807224 2.766869 3.866941 2.455648 2.274652
## 232 233 234 235 236 237 238 239
## 1 4.653852 5.135020 5.281054 5.952219 4.649091 6.018095 4.490512 2.932965
## 2 5.189298 6.314557 6.527581 6.182940 4.494536 6.049817 5.234441 6.100275
## 3 3.705792 3.899503 3.957570 3.819634 4.306913 3.896708 3.743105 4.916380
## 4 3.349510 4.826196 5.081229 4.795798 2.237307 4.663958 3.356024 4.006857
## 5 7.086045 8.097837 8.247294 7.256051 6.371138 7.184499 7.164135 7.763565
## 6 2.845373 4.094662 4.348568 4.582063 2.164414 4.571565 2.701412 2.261099
## 240 241 242 243 244 245 246 247
## 1 2.956836 3.179208 3.559831 3.174838 2.972611 4.361124 4.499702 3.449697
## 2 5.782692 5.644636 5.418724 5.676824 6.431679 5.293119 5.282827 5.730382
## 3 4.609977 4.554218 4.836874 4.812866 5.251961 5.595467 5.698876 5.306870
## 4 3.612081 3.523994 3.199172 3.505714 4.402069 3.307803 3.374542 3.724642
## 5 7.513168 7.401734 7.139957 7.340494 8.034419 6.985702 6.964082 7.498675
## 6 2.096391 2.294939 2.317889 2.233775 2.546385 2.843542 3.010876 2.298688
## 248 249 250 251 252 253 254 255
## 1 3.980918 3.426770 3.241080 3.321813 3.366958 3.347543 4.117622 2.565061
## 2 5.332551 5.535635 6.098106 6.038428 6.187201 6.732703 7.679236 7.519106
## 3 5.197006 5.027748 5.230422 5.313640 5.406880 5.560045 5.671848 6.347835
## 4 3.315715 3.504649 4.188133 4.105072 4.276291 4.932310 6.124355 5.596288
## 5 7.123772 7.327359 7.884487 7.824608 7.913195 8.440374 9.349847 8.962012
## 6 2.612063 2.227342 2.472500 2.394111 2.529107 3.107181 4.463447 4.036629
## 256 257 258 259 260 261 262 263
## 1 2.791995 2.848603 6.222798 5.518648 5.417650 5.174118 5.463188 5.971533
## 2 7.592589 8.192506 7.613422 6.131531 5.811603 5.956088 6.518475 7.320962
## 3 6.580192 5.707147 4.551395 3.894518 4.041152 3.826233 3.886648 4.317743
## 4 5.699971 6.403033 6.371304 4.637079 4.211517 4.374564 5.047654 6.017951
## 5 8.992176 9.653615 9.041679 7.629278 7.342798 7.487956 8.005527 8.762238
## 6 4.155292 4.824381 5.880265 4.531389 4.220084 4.147002 4.710711 5.556397
## 264 265 266 267 268 269 270 271
## 1 5.408871 5.364474 5.369503 5.311210 5.468341 4.684862 5.853082 4.111760
## 2 5.873985 6.084410 5.582297 5.691716 7.365150 6.850545 6.292985 5.143047
## 3 3.944671 3.834643 4.846831 4.038079 4.202225 3.938496 4.496815 4.877938
## 4 4.361095 4.543509 3.870860 4.139572 5.912661 5.158425 4.630825 2.685788
## 5 7.399951 7.595295 6.985984 7.167083 8.815364 8.370001 7.374525 6.914144
## 6 4.321892 4.369594 3.895689 4.141040 5.180260 4.218897 4.554789 2.091791
## 272 273 274 275 276 277 278 279
## 1 3.725629 3.921894 5.523364 4.973350 3.058289 4.960332 4.861229 3.070627
## 2 5.736385 5.155769 7.186009 7.238343 6.006291 7.213299 7.355393 5.795132
## 3 4.839362 4.286203 4.148237 4.380952 4.296659 4.210140 4.469765 4.503523
## 4 3.441028 2.665134 5.602045 5.598971 3.877658 5.612713 5.710319 3.612360
## 5 7.520064 7.064234 8.378912 8.386111 7.804129 8.364865 8.493915 7.564951
## 6 2.111547 1.993339 4.916068 4.705346 2.513014 4.779926 4.711343 2.285977
## 280 281 282 283 284 285 286 287
## 1 3.561388 4.216680 3.374750 5.571630 5.408034 2.345212 2.477254 2.927409
## 2 6.491419 7.105591 6.595419 8.219563 9.973467 7.928601 6.930874 7.445534
## 3 4.173615 3.975881 4.184062 4.440886 6.268257 5.936035 5.428512 6.501695
## 4 4.285849 5.167816 4.389736 6.563865 8.344298 6.072569 4.772330 5.471836
## 5 8.122015 8.703650 8.222715 9.258337 10.933893 9.363760 8.450626 8.767961
## 6 2.790991 3.947935 2.795306 5.515391 6.980733 4.422708 3.123714 4.079753
## 288 289 290 291 292 293 294 295
## 1 2.509960 2.585977 2.400932 2.683501 2.813661 2.686863 4.345410 4.479945
## 2 6.413629 6.249971 6.494620 6.930359 7.229010 6.875832 5.592884 5.181463
## 3 5.343188 5.207865 5.287981 5.569837 5.473392 5.650164 4.941731 4.796956
## 4 4.396113 4.214046 4.535171 5.074359 5.432490 5.006978 3.576391 3.058148
## 5 8.041222 7.905277 8.061694 8.582296 8.865102 8.509272 7.358111 6.995131
## 6 2.756566 2.716768 2.843575 3.668080 3.989047 3.608840 2.497209 2.387338
## 296 297 298 299 300 301 302 303
## 1 4.101880 4.215355 4.808009 2.456182 2.336450 2.394242 3.068423 2.941845
## 2 5.702683 5.379947 5.078750 7.315906 7.652372 7.121431 5.575441 5.929574
## 3 4.531097 4.405035 5.064135 5.938158 5.803548 5.507286 4.604003 4.762883
## 4 3.690081 3.281272 3.028002 5.478237 5.872746 5.284133 3.549854 3.969692
## 5 7.527753 7.235927 6.829083 8.722048 9.087048 8.634313 7.337638 7.605560
## 6 2.491650 2.393691 2.750925 3.788908 4.039227 3.734424 2.232083 2.376030
## 304 305 306 307 308 309 310 311
## 1 2.855273 3.436586 3.684046 3.978741 3.854584 4.998677 5.185967 5.719675
## 2 6.336183 6.106407 5.435975 5.842022 5.453779 4.466812 4.127252 4.624290
## 3 4.561650 4.196815 4.266337 4.017975 4.168573 4.202224 4.535816 5.594347
## 4 4.456508 4.073418 3.192729 3.845712 3.261918 2.344549 1.843049 2.847242
## 5 8.039455 7.868780 7.241481 7.658684 7.290961 6.546441 6.168368 6.187983
## 6 2.734380 2.774880 2.361712 3.039054 2.564437 2.631756 2.582365 3.293284
## 312 313 314 315 316 317 318 319
## 1 4.937184 5.460141 5.098426 5.028976 5.293601 5.460000 5.215271 4.838880
## 2 4.467670 4.023815 4.217072 4.259199 4.159874 4.029901 4.115912 4.245101
## 3 4.476328 4.594927 4.348331 4.178642 4.891978 4.889172 4.794276 4.262780
## 4 2.297145 1.850451 1.966400 2.079641 1.938234 1.959365 1.906122 1.969043
## 5 6.476538 6.069256 6.284457 6.344720 6.136907 5.966724 6.078177 6.307673
## 6 2.402724 2.904430 2.574041 2.606905 2.711944 2.970980 2.658070 2.293294
## 320 321 322 323 324 325 326 327
## 1 4.842685 4.449705 4.484974 4.585730 5.057642 4.334264 4.248227 4.277525
## 2 4.240360 4.799353 4.776401 4.757420 4.457601 5.018766 5.526219 5.218292
## 3 4.534455 4.506705 4.537503 4.811214 4.996492 4.590902 4.987239 4.873621
## 4 1.963279 2.473559 2.434921 2.406058 2.072533 2.775066 3.465771 3.042737
## 5 6.250142 6.816256 6.795014 6.731231 6.397846 6.994614 7.395759 7.139104
## 6 2.254385 1.961575 1.975895 2.005309 2.474246 1.958008 2.220200 2.005009
## 328 329 330 331 332 333 334 335
## 1 4.511237 4.336792 4.068778 4.134704 3.513382 3.127529 4.341878 4.382338
## 2 4.796609 5.372181 5.613018 5.389720 5.942590 6.090114 5.485716 5.424178
## 3 4.877258 5.105806 4.900165 4.957900 5.488342 5.279997 5.053555 5.093833
## 4 2.517471 3.305539 3.594198 3.303857 3.752625 3.884973 3.226203 3.160665
## 5 6.725423 7.068932 7.323299 7.087469 7.538669 7.661878 7.331467 7.267954
## 6 2.035908 2.350131 2.312741 2.230762 2.580536 2.403384 2.205917 2.203234
## 336 337 338 339 340 341 342 343
## 1 4.485929 4.729718 4.812037 4.629130 4.744764 4.837628 2.808348 4.362098
## 2 5.347455 5.032342 4.935566 5.124323 4.979263 4.865181 6.823411 5.448631
## 3 5.156767 5.112746 5.110110 5.016265 5.044939 4.987777 4.514441 4.784059
## 4 3.056405 2.647838 2.524309 2.778949 2.590159 2.417256 4.633530 3.331511
## 5 7.189428 6.898354 6.809633 7.000683 6.865025 6.777057 8.446366 7.165257
## 6 2.186727 2.223765 2.322145 2.157595 2.205404 2.271625 3.007511 2.617582
## 344 345 346 347 348 349 350 351
## 1 2.778316 2.366030 4.320697 4.466973 2.400876 2.213242 2.925825 3.077473
## 2 5.684615 6.353501 5.565696 5.487424 7.183736 7.319965 6.755452 6.526367
## 3 4.757585 4.927327 5.421207 5.492314 6.193135 5.934386 5.524385 5.618626
## 4 3.802233 4.548197 3.369890 3.310384 5.476084 5.437402 4.699873 4.404974
## 5 7.461637 8.044999 7.305607 7.150579 8.715944 8.820537 8.437225 8.219455
## 6 2.672462 2.940953 2.423474 2.543259 3.956148 3.799867 3.066109 2.914978
## 352 353 354 355 356 357 358 359
## 1 2.847020 3.254646 3.220675 4.018615 3.810904 8.663013 8.315674 8.225028
## 2 7.254755 7.319860 8.653858 7.691074 7.769223 4.621450 4.715344 4.749805
## 3 6.269580 6.763444 7.013329 7.521812 7.391746 6.500890 6.156007 6.172788
## 4 5.498901 5.585389 6.831393 6.079486 6.140023 5.907878 5.797393 5.786448
## 5 8.766898 8.789968 10.015464 9.117773 9.216143 5.456756 5.735313 5.758167
## 6 3.882617 4.035485 5.152518 4.846552 4.789845 6.995105 6.715094 6.630405
## 360 361 362 363 364 365 366 367
## 1 7.278021 7.215827 7.464845 7.699553 8.598542 8.542707 8.530898 7.799054
## 2 2.467536 2.805217 2.333627 2.769367 4.652258 6.207836 4.624219 2.863464
## 3 6.182375 6.025704 6.270247 6.585418 6.526351 6.398780 7.725367 6.760694
## 4 4.207340 4.344110 4.232657 4.378729 5.817697 6.961379 5.534146 4.394301
## 5 4.964540 5.192113 4.710339 4.977082 5.468166 7.258598 5.920246 4.657637
## 6 5.340790 5.361453 5.509807 5.682415 6.883975 7.392644 6.618984 5.752759
## 368 369 370 371 372 373 374 375
## 1 8.916963 7.876027 8.330037 8.374001 7.563523 8.532946 8.984585 9.736883
## 2 4.770107 4.939445 6.154059 6.280595 4.367286 5.917457 3.722610 4.767839
## 3 8.102215 6.662121 5.980344 5.981912 6.056314 6.266885 7.931335 8.798177
## 4 5.856186 5.544439 6.673537 6.801323 5.253873 6.587324 5.414656 6.304659
## 5 5.030692 6.544359 7.110024 7.285275 6.133890 6.680971 4.907474 5.412285
## 6 7.067897 6.260581 7.113702 7.203297 5.999323 7.191112 6.985946 7.809059
## 376 377 378 379 380 381 382 383
## 1 7.795568 7.924073 7.787937 8.275536 8.137870 12.821812 7.966471 8.185980
## 2 3.171317 2.549903 2.441746 3.030744 2.612679 9.894275 2.535543 2.500835
## 3 6.521129 6.722821 6.526672 7.136095 7.001182 12.116793 6.800516 7.087802
## 4 4.944969 4.652842 4.462833 5.083590 4.746519 11.163729 4.647065 4.555749
## 5 5.207964 4.521828 4.798660 4.679079 4.555128 9.663159 4.661952 4.529605
## 6 6.017927 6.001561 5.834679 6.394230 6.161471 11.755809 6.015300 6.112881
## 384 385 386 387 388 389 390 391
## 1 8.197628 9.349727 8.910330 9.193301 9.152365 8.907678 8.068366 7.677835
## 2 2.535837 4.207220 3.430304 3.965891 3.922688 3.476912 2.460378 2.173913
## 3 7.080988 8.494759 7.900222 8.245243 8.254546 7.913604 7.016902 6.584068
## 4 4.556169 5.912501 5.372451 5.774218 5.768068 5.325687 4.443905 4.162427
## 5 4.555957 4.593328 4.644399 4.813092 4.798740 4.585380 4.568138 4.613924
## 6 6.125147 7.394570 6.913418 7.236737 7.210917 6.892982 5.975128 5.602574
## 392 393 394 395 396 397 398 399
## 1 7.292592 8.487710 7.524344 7.763687 7.641959 7.677186 8.002371 9.767720
## 2 2.226899 2.983346 2.121055 2.265178 2.176869 2.197872 2.365545 4.881189
## 3 6.187448 7.518315 6.415998 6.699030 6.447121 6.503431 6.938797 8.851159
## 4 4.059557 4.886988 4.152838 4.352711 4.260815 4.238695 4.395922 6.683507
## 5 4.690817 4.584419 4.693783 4.560825 4.700448 4.730041 4.591516 5.349991
## 6 5.308181 6.426170 5.489378 5.726631 5.638596 5.653609 5.914268 7.989079
## 400 401 402 403 404 405 406 407
## 1 8.388325 8.804129 8.087911 7.825523 8.542363 9.552186 11.432606 9.054357
## 2 3.040490 3.471798 2.536220 2.216427 3.279024 4.910238 7.595875 3.994384
## 3 7.463474 7.776663 6.969235 6.627020 7.610294 8.716935 10.666205 8.166864
## 4 4.959465 5.497801 4.669753 4.375806 5.225734 6.673070 9.081383 5.594715
## 5 4.478721 4.713832 4.642624 4.544640 4.647198 5.160244 7.453981 4.877537
## 6 6.400821 6.900501 6.095533 5.820693 6.578583 7.840199 10.009989 7.043493
## 408 409 410 411 412 413 414
## 1 7.570396 7.942399 7.792350 10.375746 8.547474 9.825302 8.675535
## 2 2.751697 2.684071 3.512054 6.953620 4.401901 5.499975 4.000683
## 3 6.317911 6.868005 6.505072 9.624185 7.490223 8.978058 7.777474
## 4 4.354104 4.411296 4.984789 8.179599 5.753451 6.786214 5.592512
## 5 4.684219 4.359127 4.498617 6.248943 4.283169 4.939472 4.338544
## 6 5.605718 5.909119 6.091597 9.003486 6.898431 8.111524 6.840670
## 415 416 417 418 419 420 421
## 1 11.272810 9.378378 8.990931 9.186224 12.321200 8.708818 7.649405
## 2 6.897999 4.863694 4.682977 4.396530 8.940339 4.410853 2.275161
## 3 10.507271 8.404166 8.034930 8.321960 11.623093 7.789436 6.412841
## 4 8.419684 6.420781 6.131883 6.083046 10.281934 5.925977 4.353909
## 5 6.006137 4.335355 4.349290 4.150387 7.922425 4.248090 4.384831
## 6 9.673650 7.712449 7.339434 7.393740 11.103741 7.052418 5.696703
## 422 423 424 425 426 427 428 429
## 1 7.665560 7.356701 8.626415 8.441732 9.201143 8.265300 9.461813 8.241034
## 2 2.180745 2.508136 4.580701 4.667841 4.741929 4.569780 5.669296 3.716263
## 3 6.540748 6.403579 7.807594 7.858624 8.314551 7.743199 8.719330 7.368712
## 4 4.220529 4.151550 5.802264 5.736500 6.219861 5.681331 7.081609 5.224531
## 5 4.337641 4.332067 4.287058 4.467816 4.150667 4.462962 5.101391 3.944706
## 6 5.633333 5.359069 6.942560 6.733842 7.490506 6.578878 7.976041 6.437220
## 430 431 432 433 434 435 436 437
## 1 8.696259 7.931573 8.088359 7.540841 8.039230 8.323316 8.488298 8.855936
## 2 4.134503 3.794201 3.973307 3.782250 3.619313 3.721287 3.748221 4.457023
## 3 7.717744 7.082043 7.134695 6.790664 7.058495 7.351287 7.386206 7.874780
## 4 5.654346 5.081922 5.298499 4.904392 5.104864 5.332295 5.460661 5.994414
## 5 4.007002 4.145470 4.267862 4.462911 4.105578 3.959357 3.969580 4.147301
## 6 6.946892 6.173374 6.398175 5.806166 6.272329 6.545788 6.747217 7.180959
## 438 439 440 441 442 443 444 445
## 1 9.396533 9.435637 8.099988 8.384080 7.713189 7.607664 7.769558 8.369895
## 2 4.811409 4.695929 2.433801 2.937692 2.207332 2.192461 2.239337 2.871156
## 3 8.394236 8.484083 7.006532 7.350912 6.477458 6.351316 6.518089 7.288150
## 4 6.386947 6.327543 4.552303 5.079909 4.380978 4.219854 4.422982 4.931360
## 5 4.213632 4.342741 4.516857 4.509393 4.608229 4.755074 4.637509 3.918161
## 6 7.707796 7.700430 6.054304 6.446442 5.755888 5.609025 5.811139 6.418899
## 446 447 448 449 450 451 452 453
## 1 8.869383 7.692356 7.719083 7.662880 7.734779 8.619425 7.510000 7.420544
## 2 4.315338 2.223077 2.207460 2.126064 2.281549 4.610171 2.238968 2.096565
## 3 7.822260 6.503523 6.592742 6.535974 6.588139 7.643669 6.322861 6.315283
## 4 5.892565 4.313539 4.358067 4.262430 4.358030 5.922856 4.230132 4.095576
## 5 4.052052 4.377493 4.626360 4.619462 4.290924 4.436440 4.659548 4.698128
## 6 7.168870 5.714462 5.725384 5.655226 5.767098 7.013088 5.561251 5.419305
## 454 455 456 457 458 459 460 461
## 1 7.488797 8.615112 8.280416 8.613939 8.529698 7.413953 7.197961 7.441333
## 2 2.816811 4.549028 4.088681 4.450875 4.516870 2.377775 2.175860 2.581875
## 3 6.201144 7.628370 7.338629 7.770650 7.771267 6.463171 6.189770 6.350729
## 4 4.575975 5.918794 5.464389 5.749648 5.801327 4.266589 4.034746 4.354303
## 5 5.115979 4.321729 4.205509 4.252759 4.258883 4.236988 4.789795 4.385829
## 6 5.704355 7.016989 6.589745 6.913673 6.866815 5.484535 5.216250 5.566897
## 462 463 464 465 466 467 468 469
## 1 7.224008 7.129990 7.081032 6.761263 6.784776 8.163381 7.276556 7.067634
## 2 2.197525 2.207914 2.436963 2.400811 2.744376 4.323182 2.345760 2.548814
## 3 6.150612 6.113243 6.000826 5.995825 6.265945 7.375314 6.308918 6.388907
## 4 4.028929 4.046035 4.100801 3.940852 4.014623 5.454107 3.942868 4.149291
## 5 4.871657 4.859767 5.057480 4.982831 4.910892 4.372635 4.541440 4.648344
## 6 5.245377 5.175440 5.168781 4.825356 4.830816 6.492417 5.264318 5.133202
## 470 471 472 473 474 475 476 477
## 1 6.854434 6.816620 6.708850 6.594683 6.547902 7.501061 7.637258 7.206421
## 2 2.734118 2.311425 2.560930 2.499788 3.048901 2.434451 2.523020 2.223730
## 3 6.323801 5.996607 5.921288 5.766964 5.546301 6.666256 6.654666 6.151680
## 4 4.089454 3.742729 3.729722 3.771861 4.165060 4.053967 4.237950 3.945356
## 5 4.973919 4.971853 5.181002 5.161492 5.472082 4.527575 4.374497 4.862825
## 6 4.901491 4.816208 4.722700 4.659061 4.806139 5.415982 5.640980 5.208138
## 478 479 480 481 482 483 484 485
## 1 8.130462 7.407656 7.165587 6.328683 6.331364 6.329978 6.225603 6.407568
## 2 2.819877 2.149619 2.407795 2.841677 3.062382 3.305435 3.354835 3.007909
## 3 7.232034 6.384794 6.121775 5.803910 5.637316 5.575392 6.127800 6.133862
## 4 4.658359 4.056417 4.119396 3.815854 3.967622 4.139027 4.014627 3.945221
## 5 4.341229 4.588992 4.776866 5.387882 5.594453 5.786301 5.679794 5.329406
## 6 6.105211 5.382604 5.214375 4.428172 4.526780 4.616956 4.359692 4.497564
## 486 487 488 489 490 491 492 493
## 1 6.249662 6.708037 6.500512 7.746032 8.298286 8.670417 7.692851 7.224881
## 2 2.935648 2.355052 2.761430 3.339171 3.627365 4.033745 3.245201 3.298803
## 3 5.880484 6.022019 6.037518 6.749983 7.341242 7.743312 6.605526 6.225869
## 4 3.898792 3.748954 3.856913 4.067660 4.515760 4.922601 4.029950 3.853652
## 5 5.440584 4.972136 5.232320 5.468533 5.317298 5.364747 5.458144 5.624994
## 6 4.392934 4.727043 4.539202 5.655817 6.220009 6.633606 5.634401 5.208921
## 494 495 496 497 498 499 500 501
## 1 5.413752 5.273091 5.426681 6.011801 5.542505 5.414893 5.810165 5.688309
## 2 3.852454 3.986370 4.203245 3.717331 3.612856 3.666547 3.612745 3.521661
## 3 4.881230 4.787928 5.205773 5.359640 4.929451 4.699697 5.121802 4.887289
## 4 2.137355 2.335780 2.705197 2.322917 1.885295 1.925584 2.012924 1.880862
## 5 5.924585 6.035621 6.099197 5.645035 5.731891 5.820200 5.672489 5.683894
## 6 2.874711 2.815571 3.076628 3.570270 2.982151 2.867755 3.260115 3.150398
## 502 503 504 505 506
## 1 5.506603 5.730311 5.775209 5.732539 5.991382
## 2 4.294822 4.194855 4.555464 4.415418 4.255945
## 3 4.737543 4.960431 4.668497 4.734780 5.374542
## 4 2.309307 2.174338 2.764118 2.545516 2.378712
## 5 6.417801 6.310447 6.682914 6.547413 6.324684
## 6 3.056424 3.190441 3.518096 3.391335 3.469554
##
## $totss
## [1] 748534.8
##
## $withinss
## [1] 17974.37 26811.41 21548.41 22667.20 55771.90 25160.33
##
## $tot.withinss
## [1] 169933.6
##
## $betweenss
## [1] 578601.2myclust<-data.frame(km_bs2$cluster)
boston_scaled2$clust<-km_bs2$cluster
head(boston_scaled2)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv clust
## 1 -1.0744990 0.1595278 6
## 2 -0.4919525 -0.1014239 6
## 3 -1.2075324 1.3229375 6
## 4 -1.3601708 1.1815886 6
## 5 -1.0254866 1.4860323 6
## 6 -1.0422909 0.6705582 6lda.fit_bs2<-lda(clust~., data = boston_scaled2 )
lda.fit_bs2## Call:
## lda(clust ~ ., data = boston_scaled2)
##
## Prior probabilities of groups:
## 1 2 3 4 5 6
## 0.10079051 0.19960474 0.09486166 0.20553360 0.12845850 0.27075099
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.4149170 2.55535505 -1.228758914 -0.1951310 -1.21919439 0.78676843
## 2 0.3880377 -0.48724019 1.165421314 -0.2723291 0.98659851 -0.28553884
## 3 -0.3613809 -0.09419977 -0.474086929 1.5321752 -0.12487357 1.27068222
## 4 -0.3580718 -0.46023584 -0.003188584 -0.2723291 -0.09478548 -0.35414265
## 5 1.4172264 -0.48724019 1.069802298 0.4545202 1.34622349 -0.73713928
## 6 -0.4055840 0.02149547 -0.740804469 -0.2723291 -0.79649957 0.09099544
## age dis rad tax ptratio black
## 1 -1.4488239 1.7464736 -0.7048880 -0.5692695 -0.8353442 0.34924852
## 2 0.7651453 -0.7898745 1.1388129 1.2431405 0.6932747 0.04498348
## 3 0.2307707 -0.3386056 -0.4961654 -0.7220694 -1.1226766 0.32813467
## 4 0.4093998 -0.2612071 -0.5865335 -0.4342609 0.2608189 0.19191309
## 5 0.8557425 -0.9615698 1.2885597 1.2934457 0.4142248 -1.68787016
## 6 -0.8223904 0.7053125 -0.5694290 -0.7355910 -0.2013102 0.37698635
## lstat medv
## 1 -0.9773530 0.8760790
## 2 0.6734731 -0.5987824
## 3 -0.6138415 1.4407282
## 4 0.1508360 -0.2838601
## 5 1.1961180 -0.8078336
## 6 -0.5996059 0.2092896
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3 LD4 LD5
## crim 0.04811996 -0.28556378 -0.55488255 0.49400398 0.05329096
## zn -0.13738829 -1.83004313 0.34546140 -0.26802062 -0.87758918
## indus 0.74925386 -0.10015651 0.61607026 -0.42031079 0.25109137
## chas 0.13287282 -0.13228082 -0.94523359 -0.16829634 0.04786106
## nox 1.21764057 -0.81216848 -0.12506389 0.27633410 0.13213424
## rm -0.12060003 -0.04058521 -0.02502279 -0.75468374 0.21331834
## age 0.17397462 0.34382124 -0.07430813 -0.37956005 -0.95205471
## dis -0.36273454 -0.54652248 0.11546588 0.26210162 0.59195828
## rad 0.61453519 0.40958433 0.29006265 -0.40963042 1.56473994
## tax 0.75124298 -1.03741454 0.22707980 -0.17126395 -0.61781814
## ptratio 0.36217649 -0.18603253 0.30060517 0.16017164 -0.53729844
## black -0.27542772 0.27016025 0.77143821 -0.87012879 0.23445845
## lstat 0.48988940 -0.40861927 -0.53017288 -0.23295699 -0.06758426
## medv 0.22977036 -0.57759705 -0.86635437 -0.06977308 -0.10361245
##
## Proportion of trace:
## LD1 LD2 LD3 LD4 LD5
## 0.7285 0.1498 0.0750 0.0298 0.0168lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
plot(lda.fit_bs2, dimen = 2)
lda.arrows(lda.fit_bs2, myscale = 3)
I think the dataset could be grouped more easily by a lower number of clusters, though I’m not sure what would be the minimum number. The top three most relevant variables according to our bi-plot are “zn”, “nox”, and “tax”.
Super.Bonus
Additional ways for visualising LDA:
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutmodel_predictors <- dplyr::select(train, -crime)
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crim)k_means_matpro <- kmeans(matrix_product, centers = 6, iter.max = 10, nstart = 1, trace=FALSE)
head(train)## zn indus chas nox rm age
## 364 -0.48724019 1.0149946 3.6647712 1.8580364 -0.68548625 0.7256101
## 227 -0.48724019 -0.7196100 -0.2723291 -0.4374877 2.49832448 0.6367966
## 431 -0.48724019 1.0149946 -0.2723291 0.2528955 0.09018512 0.6225864
## 13 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.56308673 -1.0506607
## 185 -0.48724019 -1.2647715 -0.2723291 -0.5755644 -0.96871304 0.7540305
## 282 0.37030254 -1.1379558 -0.2723291 -0.9647679 0.97260025 -1.1146064
## dis rad tax ptratio black lstat
## 364 -0.8977222 1.6596029 1.5294129 0.8057784 -0.03980543 0.2782411
## 227 -0.2751294 -0.1779443 -0.6006817 -0.4875567 0.33633840 -1.3335641
## 431 -0.8274371 1.6596029 1.5294129 0.8057784 -2.99276453 0.6983467
## 13 0.7863653 -0.5224844 -0.5769480 -1.5037485 0.37051338 0.4280788
## 185 -0.3833114 -0.7521778 -1.2770905 -0.3027945 0.37599013 0.1858179
## 282 0.6884411 -0.5224844 -1.1406221 -1.6423201 0.38946296 -1.1291127
## medv crime
## 364 -0.62332733 high
## 227 1.63825412 med_high
## 431 -0.87340605 high
## 13 -0.09055093 med_low
## 185 0.42047949 med_low
## 282 1.39904839 lowmyclust <- NA
train$cl<-myclust
boston_scaled2$cl<-myclust
head(boston_scaled2)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv clust cl
## 1 -1.0744990 0.1595278 6 NA
## 2 -0.4919525 -0.1014239 6 NA
## 3 -1.2075324 1.3229375 6 NA
## 4 -1.3601708 1.1815886 6 NA
## 5 -1.0254866 1.4860323 6 NA
## 6 -1.0422909 0.6705582 6 NAhead(train)## zn indus chas nox rm age
## 364 -0.48724019 1.0149946 3.6647712 1.8580364 -0.68548625 0.7256101
## 227 -0.48724019 -0.7196100 -0.2723291 -0.4374877 2.49832448 0.6367966
## 431 -0.48724019 1.0149946 -0.2723291 0.2528955 0.09018512 0.6225864
## 13 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.56308673 -1.0506607
## 185 -0.48724019 -1.2647715 -0.2723291 -0.5755644 -0.96871304 0.7540305
## 282 0.37030254 -1.1379558 -0.2723291 -0.9647679 0.97260025 -1.1146064
## dis rad tax ptratio black lstat
## 364 -0.8977222 1.6596029 1.5294129 0.8057784 -0.03980543 0.2782411
## 227 -0.2751294 -0.1779443 -0.6006817 -0.4875567 0.33633840 -1.3335641
## 431 -0.8274371 1.6596029 1.5294129 0.8057784 -2.99276453 0.6983467
## 13 0.7863653 -0.5224844 -0.5769480 -1.5037485 0.37051338 0.4280788
## 185 -0.3833114 -0.7521778 -1.2770905 -0.3027945 0.37599013 0.1858179
## 282 0.6884411 -0.5224844 -1.1406221 -1.6423201 0.38946296 -1.1291127
## medv crime cl
## 364 -0.62332733 high NA
## 227 1.63825412 med_high NA
## 431 -0.87340605 high NA
## 13 -0.09055093 med_low NA
## 185 0.42047949 med_low NA
## 282 1.39904839 low NArownames(train)## [1] "364" "227" "431" "13" "185" "282" "491" "403" "43" "243" "446"
## [12] "273" "46" "311" "295" "371" "89" "274" "473" "75" "35" "325"
## [23] "250" "333" "204" "48" "138" "413" "14" "23" "211" "105" "251"
## [34] "398" "224" "155" "368" "447" "218" "16" "136" "285" "10" "244"
## [45] "29" "226" "493" "197" "32" "163" "97" "321" "6" "81" "466"
## [56] "280" "269" "365" "489" "187" "254" "88" "275" "450" "287" "166"
## [67] "412" "426" "149" "30" "76" "310" "394" "61" "192" "352" "399"
## [78] "257" "164" "485" "40" "372" "357" "152" "216" "208" "69" "125"
## [89] "90" "307" "288" "276" "343" "259" "484" "60" "401" "267" "99"
## [100] "147" "465" "408" "261" "84" "331" "93" "45" "461" "133" "428"
## [111] "390" "460" "500" "329" "478" "501" "270" "355" "323" "454" "37"
## [122] "58" "381" "477" "22" "198" "123" "253" "487" "283" "9" "207"
## [133] "305" "505" "416" "191" "469" "146" "302" "317" "359" "225" "451"
## [144] "496" "231" "49" "300" "113" "128" "213" "67" "238" "330" "143"
## [155] "470" "435" "332" "183" "341" "235" "98" "223" "291" "109" "91"
## [166] "289" "452" "498" "228" "95" "497" "324" "481" "17" "221" "410"
## [177] "181" "293" "137" "423" "86" "456" "444" "202" "34" "309" "326"
## [188] "482" "492" "201" "348" "506" "131" "245" "472" "230" "278" "405"
## [199] "175" "351" "12" "252" "7" "171" "360" "124" "440" "335" "3"
## [210] "220" "119" "169" "479" "344" "358" "336" "504" "170" "173" "458"
## [221] "380" "292" "400" "140" "363" "151" "83" "237" "82" "312" "248"
## [232] "26" "392" "393" "379" "129" "464" "290" "222" "50" "132" "11"
## [243] "373" "156" "475" "157" "266" "421" "180" "425" "2" "419" "160"
## [254] "369" "346" "112" "356" "322" "318" "94" "256" "5" "145" "114"
## [265] "70" "345" "241" "411" "294" "384" "306" "73" "308" "427" "21"
## [276] "168" "316" "196" "328" "319" "167" "42" "296" "375" "236" "334"
## [287] "362" "20" "120" "391" "107" "441" "194" "476" "387" "406" "51"
## [298] "249" "102" "18" "494" "420" "337" "195" "409" "366" "467" "299"
## [309] "141" "265" "92" "214" "8" "55" "28" "448" "449" "121" "179"
## [320] "315" "217" "386" "263" "407" "79" "457" "188" "142" "453" "205"
## [331] "178" "455" "135" "38" "437" "174" "52" "396" "327" "104" "101"
## [342] "502" "429" "153" "495" "71" "184" "370" "47" "117" "490" "340"
## [353] "106" "19" "165" "126" "144" "503" "395" "154" "462" "385" "240"
## [364] "111" "161" "417" "87" "85" "159" "499" "303" "279" "66" "430"
## [375] "24" "246" "247" "347" "404" "314" "103" "80" "418" "397" "353"
## [386] "339" "127" "349" "434" "468" "177" "41" "77" "63" "414" "382"
## [397] "488" "162" "74" "439" "376" "301" "436" "442"rownames(boston_scaled2)## [1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10" "11"
## [12] "12" "13" "14" "15" "16" "17" "18" "19" "20" "21" "22"
## [23] "23" "24" "25" "26" "27" "28" "29" "30" "31" "32" "33"
## [34] "34" "35" "36" "37" "38" "39" "40" "41" "42" "43" "44"
## [45] "45" "46" "47" "48" "49" "50" "51" "52" "53" "54" "55"
## [56] "56" "57" "58" "59" "60" "61" "62" "63" "64" "65" "66"
## [67] "67" "68" "69" "70" "71" "72" "73" "74" "75" "76" "77"
## [78] "78" "79" "80" "81" "82" "83" "84" "85" "86" "87" "88"
## [89] "89" "90" "91" "92" "93" "94" "95" "96" "97" "98" "99"
## [100] "100" "101" "102" "103" "104" "105" "106" "107" "108" "109" "110"
## [111] "111" "112" "113" "114" "115" "116" "117" "118" "119" "120" "121"
## [122] "122" "123" "124" "125" "126" "127" "128" "129" "130" "131" "132"
## [133] "133" "134" "135" "136" "137" "138" "139" "140" "141" "142" "143"
## [144] "144" "145" "146" "147" "148" "149" "150" "151" "152" "153" "154"
## [155] "155" "156" "157" "158" "159" "160" "161" "162" "163" "164" "165"
## [166] "166" "167" "168" "169" "170" "171" "172" "173" "174" "175" "176"
## [177] "177" "178" "179" "180" "181" "182" "183" "184" "185" "186" "187"
## [188] "188" "189" "190" "191" "192" "193" "194" "195" "196" "197" "198"
## [199] "199" "200" "201" "202" "203" "204" "205" "206" "207" "208" "209"
## [210] "210" "211" "212" "213" "214" "215" "216" "217" "218" "219" "220"
## [221] "221" "222" "223" "224" "225" "226" "227" "228" "229" "230" "231"
## [232] "232" "233" "234" "235" "236" "237" "238" "239" "240" "241" "242"
## [243] "243" "244" "245" "246" "247" "248" "249" "250" "251" "252" "253"
## [254] "254" "255" "256" "257" "258" "259" "260" "261" "262" "263" "264"
## [265] "265" "266" "267" "268" "269" "270" "271" "272" "273" "274" "275"
## [276] "276" "277" "278" "279" "280" "281" "282" "283" "284" "285" "286"
## [287] "287" "288" "289" "290" "291" "292" "293" "294" "295" "296" "297"
## [298] "298" "299" "300" "301" "302" "303" "304" "305" "306" "307" "308"
## [309] "309" "310" "311" "312" "313" "314" "315" "316" "317" "318" "319"
## [320] "320" "321" "322" "323" "324" "325" "326" "327" "328" "329" "330"
## [331] "331" "332" "333" "334" "335" "336" "337" "338" "339" "340" "341"
## [342] "342" "343" "344" "345" "346" "347" "348" "349" "350" "351" "352"
## [353] "353" "354" "355" "356" "357" "358" "359" "360" "361" "362" "363"
## [364] "364" "365" "366" "367" "368" "369" "370" "371" "372" "373" "374"
## [375] "375" "376" "377" "378" "379" "380" "381" "382" "383" "384" "385"
## [386] "386" "387" "388" "389" "390" "391" "392" "393" "394" "395" "396"
## [397] "397" "398" "399" "400" "401" "402" "403" "404" "405" "406" "407"
## [408] "408" "409" "410" "411" "412" "413" "414" "415" "416" "417" "418"
## [419] "419" "420" "421" "422" "423" "424" "425" "426" "427" "428" "429"
## [430] "430" "431" "432" "433" "434" "435" "436" "437" "438" "439" "440"
## [441] "441" "442" "443" "444" "445" "446" "447" "448" "449" "450" "451"
## [452] "452" "453" "454" "455" "456" "457" "458" "459" "460" "461" "462"
## [463] "463" "464" "465" "466" "467" "468" "469" "470" "471" "472" "473"
## [474] "474" "475" "476" "477" "478" "479" "480" "481" "482" "483" "484"
## [485] "485" "486" "487" "488" "489" "490" "491" "492" "493" "494" "495"
## [496] "496" "497" "498" "499" "500" "501" "502" "503" "504" "505" "506"train$cl <- boston_scaled2$clust[match(rownames(train), rownames(boston_scaled2))]
head(train)## zn indus chas nox rm age
## 364 -0.48724019 1.0149946 3.6647712 1.8580364 -0.68548625 0.7256101
## 227 -0.48724019 -0.7196100 -0.2723291 -0.4374877 2.49832448 0.6367966
## 431 -0.48724019 1.0149946 -0.2723291 0.2528955 0.09018512 0.6225864
## 13 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.56308673 -1.0506607
## 185 -0.48724019 -1.2647715 -0.2723291 -0.5755644 -0.96871304 0.7540305
## 282 0.37030254 -1.1379558 -0.2723291 -0.9647679 0.97260025 -1.1146064
## dis rad tax ptratio black lstat
## 364 -0.8977222 1.6596029 1.5294129 0.8057784 -0.03980543 0.2782411
## 227 -0.2751294 -0.1779443 -0.6006817 -0.4875567 0.33633840 -1.3335641
## 431 -0.8274371 1.6596029 1.5294129 0.8057784 -2.99276453 0.6983467
## 13 0.7863653 -0.5224844 -0.5769480 -1.5037485 0.37051338 0.4280788
## 185 -0.3833114 -0.7521778 -1.2770905 -0.3027945 0.37599013 0.1858179
## 282 0.6884411 -0.5224844 -1.1406221 -1.6423201 0.38946296 -1.1291127
## medv crime cl
## 364 -0.62332733 high 5
## 227 1.63825412 med_high 3
## 431 -0.87340605 high 2
## 13 -0.09055093 med_low 6
## 185 0.42047949 med_low 4
## 282 1.39904839 low 6nrow(train)## [1] 404plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type = "scatter3d", mode="markers", color = train$cl)